On the additivity of geometric invariants in Grothendieck categories

We study the additivity of various geometric invariants involved in Reimann-Roch type formulas and defined via the trace map. To do so in a general context we prove that given any Grothendieck category A, the derived category D(A) has a compatible triangulation in the sense of [May, J.P. :The Additivity of Traces in Triangulated Categories, Advances in Mathematics 163, (2001), 34-73], but not resorting to model categories. The result is proved just using the structural properties inherent to D(A). In the second part of the paper we apply compatibility to prove additivity of traces firstly and then additivity of the Chern character, interpreting this result in terms of a group homomorphism which plays the same role as the Chern character in intersection theory with the i-th Chow group replaced by the i-th Hodge cohomology group.

💡 Research Summary

The paper addresses the long‑standing problem of establishing additivity for geometric invariants that appear in Riemann‑Roch type formulas, specifically those defined via trace maps, in a setting that does not rely on model‑category machinery. The authors work in the general framework of a Grothendieck abelian category A and its derived category D(A). Their first major contribution is a proof that D(A) admits a “compatible triangulation” in the sense of May’s 2001 paper “The Additivity of Traces in Triangulated Categories.” Unlike previous approaches, which typically construct a model structure on complexes over A, choose cofibrant‑fibrant replacements, and then verify May’s axioms, the authors avoid any model‑category input. Instead, they exploit intrinsic properties of D(A): the existence of exact triangles given by cones, the shift functor, and the presence of enough dualizable objects (objects possessing strong duals). By carefully analyzing evaluation and coevaluation morphisms for these dualizable objects, they show that the trace of an endomorphism—defined as the composite ev ∘ (f ⊗ id) ∘ coev—satisfies May’s four compatibility conditions (precision, exchange, triangle, and continuity). Consequently, the trace is additive on distinguished triangles: for any triangle X → Y → Z → X